In general, \(b_1(a)\)  and \(b_2(a)\)  cannot be guaranteed to belong to the same group as \(a\) . Nevertheless, these two surrogates \(b_1(a)\)  and \(b_2(a)\)  are close enough to \(a\)  so that \(\Vert \widetilde{\Gamma } - \Gamma \Vert _{\max } \lesssim |\Gamma |_{\max }\sqrt{\log d/n}\) . This last fact and the above construction are shown in [1]}.
